Homodular pseudofunctors and bicategories of modules

Abstract

The universal property for the Bénabou bicategory of distributors (although we call them "modules") presented here is somewhat implicitly spread over a series of papers and yet, to my knowledge, does not appear in print. The inclusion of a bicategory $\mathcal{W}$ into the bicategory $\mathcal{W}\text{-}\mathrm{Mod}$ of $\mathcal{W}$-enriched categories and modules between them does have a completion property with respect to freely adjoining lax colimits (collages). Here we are interested in the universal property of the construction of $\mathcal{W}\text{-}\mathrm{Mod}$ from $\mathcal{W}\text{-}\mathrm{Cat}$. What we have in mind is an objective version of the notion of {\em homological functor} used by André Joyal in 1985.

20 pages; Section 5 is new following a question from Zeinab Galal